Time Series Modeling, Analysis and Decomposition

library('tidyverse')
library('tidyquant')
library('tseries')
library('gghighlight')
load('api_key.rdata')

Time series analysis:

• Stationary vs. Non-Stationary
• Additive vs. Exponential
• ARMA / ARIMA models
• Filters: Kalman, Holt-Winters
• Stochastic Processes, Random Walk, Wiener Process, Brownian Motion

What is a stationary time series?

“A time series is stationary if a shift in time doesn’t cause a change in the shape of the distribution. Basic properties of the distribution like the mean, variance, and covariance are constant over time.”

Stationary vs Seasonal Time series - http://www.statisticshowto.com/stationarity/

Different types of stationarity:

• Strict Stationarity - basically doesn’t exist… essentially nothing changes based off of time…
• First-Order Stationarity - means never change, but other statistics can change
• Second-Order (Weak) Stationarity - constant mean, variance, and autocovariance that doesn’t change with time
• Trend-Stationary Models - fluctuate around deterministic trend (series mean). Trend is typically linear or quadratic, amplitude of fluctuations stays constant
• Difference-Stationary Models - one or more differences of the data need to be taken to make it stationary

Why do we care about stationarity?

Most statistical models that have been developed assume stationarity.

set.seed(100)
# monthly data, normally distributed
# 100 stoltzcoin average trade price per month
# Standard deviation of 3
dates = seq(from = lubridate::ymd('2011-07-01'), 
            to = lubridate::ymd('2018-06-30'), 
            by = 'months')
n = length(dates)
mu = 100
sd = 3
time_series = tibble(date = dates, stoltzcoin = rnorm(n = n, mean = mu, sd = sd), 
                     type = 'stationary', trend = 'none')
ggplot(time_series, aes(x = date, y = stoltzcoin)) + 
  geom_line() + geom_smooth(method = lm, 
                            formula = y ~ x, se = FALSE) + 
  geom_hline(aes(yintercept = mean(stoltzcoin)), color="red")

Perform regression and look at the residuals

ts_fit = lm(stoltzcoin ~ date, data = time_series)
ts_fit

Call:
lm(formula = stoltzcoin ~ date, data = time_series)

Coefficients:
(Intercept)         date  
  1.082e+02   -5.002e-04  

res = tibble(residuals = resid(ts_fit))
ggplot(res, aes(x = 1:nrow(res), y = residuals)) + geom_point() + geom_rug(sides = 'l')

ggplot(res, aes(x = residuals)) + geom_histogram(aes(y = ..density..), bins = 40) + geom_density(col = 'red', size = 1.5)

Always test for stationarity even if it looks stationary

Augmented Dickey-Fuller is a typical test - tseries::adf.test()

• Tests null hypothesis that the autoregressive model has a unit root
• Stationarity means rejecting the null hypothesis
• We will look for p-value < 0.05

# adf.test only accepts univariate time series data, zoo object will be used
adf.test(zoo::zoo(time_series$stoltzcoin, time_series$date))

p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  zoo::zoo(time_series$stoltzcoin, time_series$date)
Dickey-Fuller = -4.1967, Lag order = 4, p-value = 0.01
alternative hypothesis: stationary

Add a linear trend for the example

Stoltzcoin has been going up by 2 each month

Known as “trend stationary”

going_up_by = 2
trend = seq(from = 0, 
            to = (n-1) * going_up_by,
            by = going_up_by)
ts_lin_trend = tibble(date = dates, stoltzcoin = time_series$stoltzcoin + trend, 
                     type = 'non-stationary', trend = 'linear')
ggplot(ts_lin_trend, aes(x = date, y = stoltzcoin)) + 
  geom_line() + geom_smooth(method = lm, 
                            formula = y ~ x, se = FALSE) + 
  geom_hline(aes(yintercept = mean(stoltzcoin)), color="red")

Perform regression and look at the residuals

ts_fit = lm(stoltzcoin ~ date, data = ts_lin_trend)
ts_fit

Call:
lm(formula = stoltzcoin ~ date, data = ts_lin_trend)

Coefficients:
(Intercept)         date  
 -887.75051      0.06521  

res = tibble(residuals = resid(ts_fit))
ggplot(res, aes(x = 1:nrow(res), y = residuals)) + geom_point() + geom_rug(sides = 'l')

ggplot(res, aes(x = residuals)) + geom_histogram(aes(y = ..density..), bins = 40) + geom_density(col = 'red', size = 1.5)

adf.test(zoo::zoo(ts_lin_trend$stoltzcoin, ts_lin_trend$date))

p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  zoo::zoo(ts_lin_trend$stoltzcoin, ts_lin_trend$date)
Dickey-Fuller = -4.1967, Lag order = 4, p-value = 0.01
alternative hypothesis: stationary

Add a seasonal component to the linear trend

Stoltzcoin fluctuates throughout the year

going_up_by = 2
trend = seq(from = 0, 
            to = (n-1) * going_up_by,
            by = going_up_by)
seasonal = sin(1:n) * 10
ts_lin_trend_seasonal = tibble(date = dates, stoltzcoin = time_series$stoltzcoin + trend + seasonal, 
                     type = 'non-stationary', trend = 'linear')
ggplot(ts_lin_trend_seasonal, aes(x = date, y = stoltzcoin)) + 
  geom_line() + geom_smooth(method = lm, 
                            formula = y ~ x, se = FALSE) + 
  geom_hline(aes(yintercept = mean(stoltzcoin)), color="red")

Perform regression and look at the residuals

ts_fit = lm(stoltzcoin ~ date, data = ts_lin_trend_seasonal)
ts_fit

Call:
lm(formula = stoltzcoin ~ date, data = ts_lin_trend_seasonal)

Coefficients:
(Intercept)         date  
 -887.83081      0.06523  

res = tibble(residuals = resid(ts_fit))
ggplot(res, aes(x = 1:nrow(res), y = residuals)) + geom_point() + geom_rug(sides = 'l')

ggplot(res, aes(x = residuals)) + geom_histogram(aes(y = ..density..), bins = 40) + geom_density(col = 'red', size = 1.5)

adf.test(zoo::zoo(ts_lin_trend_seasonal$stoltzcoin, ts_lin_trend_seasonal$date))

p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  zoo::zoo(ts_lin_trend_seasonal$stoltzcoin, ts_lin_trend_seasonal$date)
Dickey-Fuller = -6.4213, Lag order = 4, p-value = 0.01
alternative hypothesis: stationary

Add an exponential trend to the seasonal component

Stoltzcoin has increased in volatility over the years

• Stoltzcoin now has:
  • A positive and growing linear trend
  • Seasonality

going_up_by = 2
trend = seq(from = 0, 
            to = (n-1) * going_up_by,
            by = going_up_by)
seasonal = sin(1:n) * 1:n
ts_lin_trend_seasonal_exp = tibble(date = dates, stoltzcoin = time_series$stoltzcoin + trend + seasonal, 
                     type = 'non-stationary', trend = 'linear')
ggplot(ts_lin_trend_seasonal_exp, aes(x = date, y = stoltzcoin)) + 
  geom_line() + geom_smooth(method = lm, 
                            formula = y ~ x, se = FALSE) + 
  geom_hline(aes(yintercept = mean(stoltzcoin)), color="red")

Perform regression and look at the residuals

ts_fit = lm(stoltzcoin ~ date, data = ts_lin_trend_seasonal_exp)
ts_fit

Call:
lm(formula = stoltzcoin ~ date, data = ts_lin_trend_seasonal_exp)

Coefficients:
(Intercept)         date  
 -925.35762      0.06756  

res = tibble(residuals = resid(ts_fit))
ggplot(res, aes(x = 1:nrow(res), y = residuals)) + geom_point() + geom_rug(sides = 'l')

ggplot(res, aes(x = residuals)) + geom_histogram(aes(y = ..density..), bins = 40) + geom_density(col = 'red', size = 1.5)

adf.test(zoo::zoo(ts_lin_trend_seasonal_exp$stoltzcoin, ts_lin_trend_seasonal_exp$date))

p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  zoo::zoo(ts_lin_trend_seasonal_exp$stoltzcoin, ts_lin_trend_seasonal_exp$date)
Dickey-Fuller = -6.2925, Lag order = 4, p-value = 0.01
alternative hypothesis: stationary

R’s decompose function does a lot for you

d1y =  as.numeric(lubridate::year(min(ts_lin_trend_seasonal_exp$date)))
d1m = as.numeric(lubridate::month(min(ts_lin_trend_seasonal_exp$date)))
d2y =  as.numeric(lubridate::year(max(ts_lin_trend_seasonal_exp$date)))
d2m = as.numeric(lubridate::month(max(ts_lin_trend_seasonal_exp$date)))
x = ts(ts_lin_trend_seasonal_exp$stoltzcoin, start=c(d1y, d1m), end=c(d2y, d2m), frequency=12)
plot(decompose(x, type = 'additive'))

plot(decompose(x, type = 'multiplicative'))

Example from decompose function

random portion should just look like white noise (otherwise you’ve likely missed something)

require(graphics)
m <- decompose(co2)
plot(m)

dow = tq_index("DOW") %>%
  tq_get(get = "stock.prices")
head(dow, 10)

p = ggplot(dow, aes(x = date, y = close, col = symbol)) + 
  geom_line() + 
  gghighlight(max(close) > 250) + 
  theme_minimal() + 
  facet_wrap(~ symbol)
print(p)

Is BA a stationary time series?

Can you simply zoom in and find a portion that is stationary?

ba = dow %>% 
  filter(symbol == 'BA') %>% 
  select(date, close)
plotly::ggplotly(ggplot(ba, aes(x = date, y = close)) + geom_line() + geom_smooth(method = lm, se = FALSE))

ts_fit

Call:
lm(formula = close ~ date, data = ba)

Coefficients:
(Intercept)         date  
 -734.75085      0.05385  

Are the daily returns stationary?

ret = ba %>%
  tq_transmute(close, mutate_fun = dailyReturn)
ggplot(ret, aes(x = date, y = daily.returns)) + geom_line() + geom_smooth(method = lm, se = FALSE)

adf.test(zoo::zoo(ret$daily.returns, ret$date))

p-value smaller than printed p-value

    Augmented Dickey-Fuller Test

data:  zoo::zoo(ret$daily.returns, ret$date)
Dickey-Fuller = -14.308, Lag order = 13, p-value = 0.01
alternative hypothesis: stationary

ts_fit = lm(daily.returns ~ date, data = ret)
ts_fit

Call:
lm(formula = daily.returns ~ date, data = ret)

Coefficients:
(Intercept)         date  
 -8.376e-03    5.736e-07  

res = tibble(residuals = resid(ts_fit))
ggplot(res, aes(x = 1:nrow(res), y = residuals)) + geom_point() + geom_rug(sides = 'l')

ggplot(res, aes(x = residuals)) + geom_histogram(aes(y = ..density..), bins = 40) + geom_density(col = 'red', size = 1.5)